Tischler graphs of critically fixed rational maps and their applications

Hlushchanka, Mikhail

doi:10.48550/arxiv.1904.04759

Cited by 5 publications

(10 citation statements)

References 9 publications

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“…In order to prepare for the proof, we will have a closer look at the pull-back action of Schottky maps on simple closed curves in S 2 \ P . Some of the following parallels the concepts and arguments in the orientation-preserving case in [Hlu19]. As in Section 3, we will adopt the convention that homotopy of simple closed curves will mean free homotopy in S 2 \ P .…”

Section: Plane Graphs and Schottky Mapsmentioning

confidence: 99%

“…We should remark on one essential difference between the orientation-preserving and orientationreversing cases. In the orientation-preserving version of Lemma 5.7, strict inequality always holds if γ intersects an edge of T more than once [Hlu19]. As a consequence, a critically fixed rational maps f always has a global curve attractor A(f ), which is a finite set of homotopy classes of simple closed curves in Ĉ \ P f such that for every simple closed curve γ ⊂ Ĉ \ P f there exists N such that for all n ≥ N , all components of f −n (γ) are contained in A(f ).…”

Section: Plane Graphs and Schottky Mapsmentioning

confidence: 99%

“…The main result in this paper is the classification of critically fixed anti-rational maps via certain plane graphs. The analogous problem for critically fixed rational maps was solved by Hlushchanka [Hlu19], following partial results in [CGN + 15]. A classification of critically fixed anti-polynomials with simple critical points was obtained in [Gey08] for the case of real coefficients, and in [LMM19] for general complex coefficients.…”

Section: Introductionmentioning

confidence: 99%

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Classification of critically fixed anti-rational maps

Geyer¹

2020

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We show that there is a one-to-one correspondence between conjugacy classes of critically fixed anti-rational maps and equivalence classes of certain plane graphs. We furthermore prove that critically fixed anti-rational maps are Thurston equivalent to "topological Schottky maps" associated to these plane graphs, given in each face by a topological reflection in its boundary. As a corollary, we obtain a similar classification of critically fixed anti-polynomials by certain plane trees. One of the main technical tools is an analogue of Thurston's characterization of rational maps in the orientation-reversing case. Contents 1. Introduction 1 2. Review of anti-holomorphic dynamics 2 3. Thurston's theorem for orientation-reversing maps 4 4. Critically fixed anti-rational maps and Tischler graphs 10 5. Plane graphs and Schottky maps 14 6. Critically fixed anti-polynomials 21 7. Examples and Applications 22 References 26

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Section: Plane Graphs and Schottky Mapsmentioning

confidence: 99%

Section: Plane Graphs and Schottky Mapsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Classification of critically fixed anti-rational maps

Geyer¹

2020

Preprint

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“…‚ Jordan curves [BM17] and Trees [Hlu17] of expanding Thurston maps. ‚ Critically fixed rational maps [CGN `15], [Hlu19]. ‚ Jordan curves [GHMZ18] and trees [Hlu17] of Sierpinsky Carpet rational map.…”

Section: Finite Subdivision Rulesmentioning

confidence: 99%

Levy and Thurston obstructions of finite subdivision rules

Park¹

2020

Preprint

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For a post-critically finite branched covering of the sphere that is a subdivision map of a finite subdivision rule, we define non-expanding spines which determine the existence of a Levy cycle in a non-exhaustive semi-decidable algorithm. Especially when a finite subdivision rule has polynomial growth of edge subdivisions, the algorithm terminates very quickly, and the existence of a Levy cycle is equivalent to the existence of a Thurston obstruction. In order to show the equivalence between Levy and Thurston obstructions, we generalize the arcs intersecting obstruction theorem by Pilgrim and Tan to a graph intersecting obstruction theorem. As a corollary, we prove that for a pair of post-critically finite polynomials, if at least one polynomial has core entropy zero, then their mating has a Levy cycle if and only if the mating has a Thurston obstruction.

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“…Recently, Belk-Lanier-Margalit-Winarski proved the existence of a finite global curve attractor for all postcritically-finite polynomials [BLMW19]. The conjecture is also known to be true for all critically fixed rational maps (that is, rational maps for which each critical point is fixed) and some nearly Euclidean Thurston maps (that is, Thurston maps with exactly four postcritical points and only simple critical points); see [Hlu19] and [Lod13, FKK + 17]. In [KL19] Gregory Kelsey and Russell Lodge verified the conjecture for all quadratic non-Lattès maps with four postcritical points.…”

Section: Introductionmentioning

confidence: 99%

Eliminating Thurston obstructions and controlling dynamics on curves

Bonk,

Hlushchanka,

Iseli

2021

Preprint

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Every Thurston map f ∶ S 2 → S 2 on a 2-sphere S 2 induces a pull-back operation on Jordan curves α ⊂ S 2 ∖ P f , where P f is the postcritical set of f . Here the isotopy class [f −1 (α)] (relative to P f ) only depends on the isotopy class [α]. We study this operation for Thurston maps with four postcritical points. In this case a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation.We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can "blow up" suitable arcs in the underlying 2-sphere and construct a new Thurston map f for which this obstruction is eliminated. We prove that no other obstruction arises and so f is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points.We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

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Tischler graphs of critically fixed rational maps and their applications

Cited by 5 publications

References 9 publications

Classification of critically fixed anti-rational maps

Classification of critically fixed anti-rational maps

Levy and Thurston obstructions of finite subdivision rules

Eliminating Thurston obstructions and controlling dynamics on curves

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